Gaussian Elimination

In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss, but it was not invented by him.

Elementary row operations are used to reduce a matrix to what is called triangular form (in numerical analysis) or row echelon form (in abstract algebra). Gauss–Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form. Gaussian elimination alone is sufficient for solving systems of linear equations in many applications, and requires fewer calculations than the Gauss–Jordan version.

Read more about Gaussian EliminationHistory, Algorithm Overview, Example, Analysis, Pseudocode

Other articles related to "gaussian elimination, elimination, gaussian":

Matrix Decomposition - Decompositions Related To Solving Systems of Linear Equations - LU Decomposition
... decompositions summarize the process of Gaussian elimination in matrix form ... Matrix P represents any row interchanges carried out in the process of Gaussian elimination ... If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P=I, so an LU decomposition exists ...
Pivot Element - Examples of Systems That Require Pivoting
... In the case of Gaussian elimination, the algorithm requires that pivot elements not be zero ... The following system is dramatically affected by round-off error when Gaussian elimination and backwards substitution are performed This system has the exact solution of x1 = 10.00 and x2 = 1.000, but when the ... so that a21 is in the pivot position Considering this system, the elimination algorithm and backwards substitution using four-digit arithmetic yield the correct values x1 = 10.0 ...
Gaussian Elimination - Pseudocode
... As explained above, Gaussian elimination writes a given m × n matrix A uniquely as a product of an invertible m × m matrix S and a row-echelon matrix T ... programmers now exploit thread-level parallel Gaussian elimination algorithms to increase the speed of computing ...
Optical Aberration - Chromatic or Color Aberration
... monochromatic aberrations be neglected — in other words, the Gaussian theory be accepted — then every reproduction is determined by the positions of the ... of reproduction be achromatized, then the Gaussian image for all distances of objects is the same for the two colors, and the system is said to be in stable achromatism ... Two other conditions may also be postulated one is always the elimination of the aberration on the axis the second either the Herschel or Fraunhofer Condition, the latter being the best vide supra ...
LU Decomposition - Applications - Solving Linear Equations
... forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself) ... for the different b, rather than using Gaussian elimination each time ... The matrices L and U could be thought to have "encoded" the Gaussian elimination process ...

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